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Theorem infdifsn 9655
Description: Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
infdifsn (Ο‰ β‰Ό 𝐴 β†’ (𝐴 βˆ– {𝐡}) β‰ˆ 𝐴)
Proof of Theorem infdifsn
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 brdomi 8957 . . . 4 (Ο‰ β‰Ό 𝐴 β†’ βˆƒπ‘“ 𝑓:ω–1-1→𝐴)
21adantr 480 . . 3 ((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ βˆƒπ‘“ 𝑓:ω–1-1→𝐴)
3 reldom 8948 . . . . . . 7 Rel β‰Ό
43brrelex2i 5733 . . . . . 6 (Ο‰ β‰Ό 𝐴 β†’ 𝐴 ∈ V)
54ad2antrr 723 . . . . 5 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ 𝐴 ∈ V)
6 simplr 766 . . . . 5 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ 𝐡 ∈ 𝐴)
7 f1f 6787 . . . . . . 7 (𝑓:ω–1-1→𝐴 β†’ 𝑓:Ο‰βŸΆπ΄)
87adantl 481 . . . . . 6 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ 𝑓:Ο‰βŸΆπ΄)
9 peano1 7882 . . . . . 6 βˆ… ∈ Ο‰
10 ffvelcdm 7083 . . . . . 6 ((𝑓:Ο‰βŸΆπ΄ ∧ βˆ… ∈ Ο‰) β†’ (π‘“β€˜βˆ…) ∈ 𝐴)
118, 9, 10sylancl 585 . . . . 5 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (π‘“β€˜βˆ…) ∈ 𝐴)
12 difsnen 9056 . . . . 5 ((𝐴 ∈ V ∧ 𝐡 ∈ 𝐴 ∧ (π‘“β€˜βˆ…) ∈ 𝐴) β†’ (𝐴 βˆ– {𝐡}) β‰ˆ (𝐴 βˆ– {(π‘“β€˜βˆ…)}))
135, 6, 11, 12syl3anc 1370 . . . 4 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (𝐴 βˆ– {𝐡}) β‰ˆ (𝐴 βˆ– {(π‘“β€˜βˆ…)}))
14 vex 3477 . . . . . . . . . 10 𝑓 ∈ V
15 f1f1orn 6844 . . . . . . . . . . 11 (𝑓:ω–1-1→𝐴 β†’ 𝑓:ω–1-1-ontoβ†’ran 𝑓)
1615adantl 481 . . . . . . . . . 10 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ 𝑓:ω–1-1-ontoβ†’ran 𝑓)
17 f1oen3g 8965 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:ω–1-1-ontoβ†’ran 𝑓) β†’ Ο‰ β‰ˆ ran 𝑓)
1814, 16, 17sylancr 586 . . . . . . . . 9 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ Ο‰ β‰ˆ ran 𝑓)
1918ensymd 9004 . . . . . . . 8 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ ran 𝑓 β‰ˆ Ο‰)
203brrelex1i 5732 . . . . . . . . . . 11 (Ο‰ β‰Ό 𝐴 β†’ Ο‰ ∈ V)
2120ad2antrr 723 . . . . . . . . . 10 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ Ο‰ ∈ V)
22 limom 7874 . . . . . . . . . . 11 Lim Ο‰
2322limenpsi 9155 . . . . . . . . . 10 (Ο‰ ∈ V β†’ Ο‰ β‰ˆ (Ο‰ βˆ– {βˆ…}))
2421, 23syl 17 . . . . . . . . 9 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ Ο‰ β‰ˆ (Ο‰ βˆ– {βˆ…}))
2514resex 6029 . . . . . . . . . . 11 (𝑓 β†Ύ (Ο‰ βˆ– {βˆ…})) ∈ V
26 simpr 484 . . . . . . . . . . . 12 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ 𝑓:ω–1-1→𝐴)
27 difss 4131 . . . . . . . . . . . 12 (Ο‰ βˆ– {βˆ…}) βŠ† Ο‰
28 f1ores 6847 . . . . . . . . . . . 12 ((𝑓:ω–1-1→𝐴 ∧ (Ο‰ βˆ– {βˆ…}) βŠ† Ο‰) β†’ (𝑓 β†Ύ (Ο‰ βˆ– {βˆ…})):(Ο‰ βˆ– {βˆ…})–1-1-ontoβ†’(𝑓 β€œ (Ο‰ βˆ– {βˆ…})))
2926, 27, 28sylancl 585 . . . . . . . . . . 11 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (𝑓 β†Ύ (Ο‰ βˆ– {βˆ…})):(Ο‰ βˆ– {βˆ…})–1-1-ontoβ†’(𝑓 β€œ (Ο‰ βˆ– {βˆ…})))
30 f1oen3g 8965 . . . . . . . . . . 11 (((𝑓 β†Ύ (Ο‰ βˆ– {βˆ…})) ∈ V ∧ (𝑓 β†Ύ (Ο‰ βˆ– {βˆ…})):(Ο‰ βˆ– {βˆ…})–1-1-ontoβ†’(𝑓 β€œ (Ο‰ βˆ– {βˆ…}))) β†’ (Ο‰ βˆ– {βˆ…}) β‰ˆ (𝑓 β€œ (Ο‰ βˆ– {βˆ…})))
3125, 29, 30sylancr 586 . . . . . . . . . 10 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (Ο‰ βˆ– {βˆ…}) β‰ˆ (𝑓 β€œ (Ο‰ βˆ– {βˆ…})))
32 f1orn 6843 . . . . . . . . . . . . 13 (𝑓:ω–1-1-ontoβ†’ran 𝑓 ↔ (𝑓 Fn Ο‰ ∧ Fun ◑𝑓))
3332simprbi 496 . . . . . . . . . . . 12 (𝑓:ω–1-1-ontoβ†’ran 𝑓 β†’ Fun ◑𝑓)
34 imadif 6632 . . . . . . . . . . . 12 (Fun ◑𝑓 β†’ (𝑓 β€œ (Ο‰ βˆ– {βˆ…})) = ((𝑓 β€œ Ο‰) βˆ– (𝑓 β€œ {βˆ…})))
3516, 33, 343syl 18 . . . . . . . . . . 11 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (𝑓 β€œ (Ο‰ βˆ– {βˆ…})) = ((𝑓 β€œ Ο‰) βˆ– (𝑓 β€œ {βˆ…})))
36 f1fn 6788 . . . . . . . . . . . . . 14 (𝑓:ω–1-1→𝐴 β†’ 𝑓 Fn Ο‰)
3736adantl 481 . . . . . . . . . . . . 13 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ 𝑓 Fn Ο‰)
38 fnima 6680 . . . . . . . . . . . . 13 (𝑓 Fn Ο‰ β†’ (𝑓 β€œ Ο‰) = ran 𝑓)
3937, 38syl 17 . . . . . . . . . . . 12 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (𝑓 β€œ Ο‰) = ran 𝑓)
40 fnsnfv 6970 . . . . . . . . . . . . . 14 ((𝑓 Fn Ο‰ ∧ βˆ… ∈ Ο‰) β†’ {(π‘“β€˜βˆ…)} = (𝑓 β€œ {βˆ…}))
4137, 9, 40sylancl 585 . . . . . . . . . . . . 13 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ {(π‘“β€˜βˆ…)} = (𝑓 β€œ {βˆ…}))
4241eqcomd 2737 . . . . . . . . . . . 12 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (𝑓 β€œ {βˆ…}) = {(π‘“β€˜βˆ…)})
4339, 42difeq12d 4123 . . . . . . . . . . 11 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ ((𝑓 β€œ Ο‰) βˆ– (𝑓 β€œ {βˆ…})) = (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}))
4435, 43eqtrd 2771 . . . . . . . . . 10 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (𝑓 β€œ (Ο‰ βˆ– {βˆ…})) = (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}))
4531, 44breqtrd 5174 . . . . . . . . 9 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (Ο‰ βˆ– {βˆ…}) β‰ˆ (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}))
46 entr 9005 . . . . . . . . 9 ((Ο‰ β‰ˆ (Ο‰ βˆ– {βˆ…}) ∧ (Ο‰ βˆ– {βˆ…}) β‰ˆ (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)})) β†’ Ο‰ β‰ˆ (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}))
4724, 45, 46syl2anc 583 . . . . . . . 8 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ Ο‰ β‰ˆ (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}))
48 entr 9005 . . . . . . . 8 ((ran 𝑓 β‰ˆ Ο‰ ∧ Ο‰ β‰ˆ (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)})) β†’ ran 𝑓 β‰ˆ (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}))
4919, 47, 48syl2anc 583 . . . . . . 7 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ ran 𝑓 β‰ˆ (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}))
50 difexg 5327 . . . . . . . 8 (𝐴 ∈ V β†’ (𝐴 βˆ– ran 𝑓) ∈ V)
51 enrefg 8983 . . . . . . . 8 ((𝐴 βˆ– ran 𝑓) ∈ V β†’ (𝐴 βˆ– ran 𝑓) β‰ˆ (𝐴 βˆ– ran 𝑓))
525, 50, 513syl 18 . . . . . . 7 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (𝐴 βˆ– ran 𝑓) β‰ˆ (𝐴 βˆ– ran 𝑓))
53 disjdif 4471 . . . . . . . 8 (ran 𝑓 ∩ (𝐴 βˆ– ran 𝑓)) = βˆ…
5453a1i 11 . . . . . . 7 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (ran 𝑓 ∩ (𝐴 βˆ– ran 𝑓)) = βˆ…)
55 difss 4131 . . . . . . . . . 10 (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) βŠ† ran 𝑓
56 ssrin 4233 . . . . . . . . . 10 ((ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) βŠ† ran 𝑓 β†’ ((ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) ∩ (𝐴 βˆ– ran 𝑓)) βŠ† (ran 𝑓 ∩ (𝐴 βˆ– ran 𝑓)))
5755, 56ax-mp 5 . . . . . . . . 9 ((ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) ∩ (𝐴 βˆ– ran 𝑓)) βŠ† (ran 𝑓 ∩ (𝐴 βˆ– ran 𝑓))
58 sseq0 4399 . . . . . . . . 9 ((((ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) ∩ (𝐴 βˆ– ran 𝑓)) βŠ† (ran 𝑓 ∩ (𝐴 βˆ– ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐴 βˆ– ran 𝑓)) = βˆ…) β†’ ((ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) ∩ (𝐴 βˆ– ran 𝑓)) = βˆ…)
5957, 53, 58mp2an 689 . . . . . . . 8 ((ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) ∩ (𝐴 βˆ– ran 𝑓)) = βˆ…
6059a1i 11 . . . . . . 7 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ ((ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) ∩ (𝐴 βˆ– ran 𝑓)) = βˆ…)
61 unen 9049 . . . . . . 7 (((ran 𝑓 β‰ˆ (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) ∧ (𝐴 βˆ– ran 𝑓) β‰ˆ (𝐴 βˆ– ran 𝑓)) ∧ ((ran 𝑓 ∩ (𝐴 βˆ– ran 𝑓)) = βˆ… ∧ ((ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) ∩ (𝐴 βˆ– ran 𝑓)) = βˆ…)) β†’ (ran 𝑓 βˆͺ (𝐴 βˆ– ran 𝑓)) β‰ˆ ((ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) βˆͺ (𝐴 βˆ– ran 𝑓)))
6249, 52, 54, 60, 61syl22anc 836 . . . . . 6 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (ran 𝑓 βˆͺ (𝐴 βˆ– ran 𝑓)) β‰ˆ ((ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) βˆͺ (𝐴 βˆ– ran 𝑓)))
638frnd 6725 . . . . . . 7 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ ran 𝑓 βŠ† 𝐴)
64 undif 4481 . . . . . . 7 (ran 𝑓 βŠ† 𝐴 ↔ (ran 𝑓 βˆͺ (𝐴 βˆ– ran 𝑓)) = 𝐴)
6563, 64sylib 217 . . . . . 6 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (ran 𝑓 βˆͺ (𝐴 βˆ– ran 𝑓)) = 𝐴)
66 uncom 4153 . . . . . . 7 ((ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) βˆͺ (𝐴 βˆ– ran 𝑓)) = ((𝐴 βˆ– ran 𝑓) βˆͺ (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}))
67 eldifn 4127 . . . . . . . . . . 11 ((π‘“β€˜βˆ…) ∈ (𝐴 βˆ– ran 𝑓) β†’ Β¬ (π‘“β€˜βˆ…) ∈ ran 𝑓)
68 fnfvelrn 7082 . . . . . . . . . . . 12 ((𝑓 Fn Ο‰ ∧ βˆ… ∈ Ο‰) β†’ (π‘“β€˜βˆ…) ∈ ran 𝑓)
6937, 9, 68sylancl 585 . . . . . . . . . . 11 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (π‘“β€˜βˆ…) ∈ ran 𝑓)
7067, 69nsyl3 138 . . . . . . . . . 10 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ Β¬ (π‘“β€˜βˆ…) ∈ (𝐴 βˆ– ran 𝑓))
71 disjsn 4715 . . . . . . . . . 10 (((𝐴 βˆ– ran 𝑓) ∩ {(π‘“β€˜βˆ…)}) = βˆ… ↔ Β¬ (π‘“β€˜βˆ…) ∈ (𝐴 βˆ– ran 𝑓))
7270, 71sylibr 233 . . . . . . . . 9 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ ((𝐴 βˆ– ran 𝑓) ∩ {(π‘“β€˜βˆ…)}) = βˆ…)
73 undif4 4466 . . . . . . . . 9 (((𝐴 βˆ– ran 𝑓) ∩ {(π‘“β€˜βˆ…)}) = βˆ… β†’ ((𝐴 βˆ– ran 𝑓) βˆͺ (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)})) = (((𝐴 βˆ– ran 𝑓) βˆͺ ran 𝑓) βˆ– {(π‘“β€˜βˆ…)}))
7472, 73syl 17 . . . . . . . 8 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ ((𝐴 βˆ– ran 𝑓) βˆͺ (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)})) = (((𝐴 βˆ– ran 𝑓) βˆͺ ran 𝑓) βˆ– {(π‘“β€˜βˆ…)}))
75 uncom 4153 . . . . . . . . . 10 ((𝐴 βˆ– ran 𝑓) βˆͺ ran 𝑓) = (ran 𝑓 βˆͺ (𝐴 βˆ– ran 𝑓))
7675, 65eqtrid 2783 . . . . . . . . 9 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ ((𝐴 βˆ– ran 𝑓) βˆͺ ran 𝑓) = 𝐴)
7776difeq1d 4121 . . . . . . . 8 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (((𝐴 βˆ– ran 𝑓) βˆͺ ran 𝑓) βˆ– {(π‘“β€˜βˆ…)}) = (𝐴 βˆ– {(π‘“β€˜βˆ…)}))
7874, 77eqtrd 2771 . . . . . . 7 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ ((𝐴 βˆ– ran 𝑓) βˆͺ (ran 𝑓 βˆ– {(π‘“β€˜βˆ…)})) = (𝐴 βˆ– {(π‘“β€˜βˆ…)}))
7966, 78eqtrid 2783 . . . . . 6 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ ((ran 𝑓 βˆ– {(π‘“β€˜βˆ…)}) βˆͺ (𝐴 βˆ– ran 𝑓)) = (𝐴 βˆ– {(π‘“β€˜βˆ…)}))
8062, 65, 793brtr3d 5179 . . . . 5 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ 𝐴 β‰ˆ (𝐴 βˆ– {(π‘“β€˜βˆ…)}))
8180ensymd 9004 . . . 4 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (𝐴 βˆ– {(π‘“β€˜βˆ…)}) β‰ˆ 𝐴)
82 entr 9005 . . . 4 (((𝐴 βˆ– {𝐡}) β‰ˆ (𝐴 βˆ– {(π‘“β€˜βˆ…)}) ∧ (𝐴 βˆ– {(π‘“β€˜βˆ…)}) β‰ˆ 𝐴) β†’ (𝐴 βˆ– {𝐡}) β‰ˆ 𝐴)
8313, 81, 82syl2anc 583 . . 3 (((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) β†’ (𝐴 βˆ– {𝐡}) β‰ˆ 𝐴)
842, 83exlimddv 1937 . 2 ((Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ (𝐴 βˆ– {𝐡}) β‰ˆ 𝐴)
85 difsn 4801 . . . 4 (Β¬ 𝐡 ∈ 𝐴 β†’ (𝐴 βˆ– {𝐡}) = 𝐴)
8685adantl 481 . . 3 ((Ο‰ β‰Ό 𝐴 ∧ Β¬ 𝐡 ∈ 𝐴) β†’ (𝐴 βˆ– {𝐡}) = 𝐴)
87 enrefg 8983 . . . . 5 (𝐴 ∈ V β†’ 𝐴 β‰ˆ 𝐴)
884, 87syl 17 . . . 4 (Ο‰ β‰Ό 𝐴 β†’ 𝐴 β‰ˆ 𝐴)
8988adantr 480 . . 3 ((Ο‰ β‰Ό 𝐴 ∧ Β¬ 𝐡 ∈ 𝐴) β†’ 𝐴 β‰ˆ 𝐴)
9086, 89eqbrtrd 5170 . 2 ((Ο‰ β‰Ό 𝐴 ∧ Β¬ 𝐡 ∈ 𝐴) β†’ (𝐴 βˆ– {𝐡}) β‰ˆ 𝐴)
9184, 90pm2.61dan 810 1 (Ο‰ β‰Ό 𝐴 β†’ (𝐴 βˆ– {𝐡}) β‰ˆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1540  βˆƒwex 1780   ∈ wcel 2105  Vcvv 3473   βˆ– cdif 3945   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628   class class class wbr 5148  β—‘ccnv 5675  ran crn 5677   β†Ύ cres 5678   β€œ cima 5679  Fun wfun 6537   Fn wfn 6538  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  Ο‰com 7858   β‰ˆ cen 8939   β‰Ό cdom 8940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7859  df-er 8706  df-en 8943  df-dom 8944
This theorem is referenced by:  infdiffi  9656  infdju1  10187  infpss  10215
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